Regardless of which study group one is put in, all patients will take afatinib by mouth every day. The first cycle will last 28 days. All cycles after that will last 25-31 days. Patients will take their medication (tablets) by mouth once a day, at about the same time each day. They should take afatinib with a glass of water. Afatinb treatment will continue until the assigned course is completed, or until there are side effects that cannot be tolerated, or one decides to stop study treatment, of if the lung cancer returns.